Geometry of the 3-Qubit State, Entanglement and Division Algebras

TL;DR

This paper extends geometric representations of qubits to three-qubit systems using higher-dimensional spheres and Hopf fibrations, revealing entanglement-sensitive structures and proposing measures for entanglement degree.

Contribution

It introduces a geometric framework for 3-qubit states based on 15-sphere representations and Hopf fibrations, generalizing previous 1- and 2-qubit models.

Findings

Representation of 3-qubit states on a 15-sphere

Entanglement sensitivity of the geometric map

Definition of a measure for 3-qubit entanglement

Abstract

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere $S^{15}$, which allows for a natural (last) Hopf fibration with $S^8$ as base and $S^7$ as fiber. A striking feature is, as in the case of 1 and 2 qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling 3 qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the 3-qubit state. Conjectures on the possibility to generalize the construction for higher qubit states are also discussed.